The present invention provides a computationally efficient method and an associated apparatus for computing the output signal g of a Linear Shift-Variant System (LSVS) given the input signal f and a set of moment parameters w of the Shift-Variant Point Spread Function (SV-PSF) h that characterizes the LSVS. The present invention also provides a computationally efficient method and an associated apparatus for computing the input signal or restored signal f of an LSVS given the output signal g and a set of moment parameters w of the SV-PSF h that characterizes the LSVS. This invention is based on a new signal processing transform named Rao Transform (RT) which was invented recently by the author of the present invention and first disclosed here. Forward RT is used for computing the output signal from input signal and inverse RT is used for computing the input or restored signal from the output signal.
There are many physical systems that can be modeled by an LSVS. A typical example is a misfocused camera system where the input signal is a focused image f, output signal g is the blurred image recorded by the camera, and an SV-PSF h of the camera which completely characterizes the image formation in the camera. We use a misfocused camera as an important example for describing the methods and apparatus of the present invention for the sake of simplicity, clarity, and brevity. However, the present invention is not limited to this particular example alone, but is applicable to any general LSVS.
In the case of a misfocused camera, the present invention can be used to restore shift-variant blurred images recorded by the camera given a set of moment parameters w of its SV-PSF h. This application is sometimes referred to as deblurring or restoration of spatially-variant blurred images or restoration of shift-variant defocused images. The present invention can also be used to efficiently compute the output blurred image g of the camera given the input focused image fand the moments of the derivatives of the SV-PSF h. This problem of computing the output signal g from the input signal f arises in the computer simulation and design of a camera system. It may also be used in generating computer animated movie frames where an image frame should contain shift-variant blur to produce the effect of depth or 3D in a scene.
The present invention is based on the recently invented RT for signal processing. RT is a more efficient and natural alternative to the conventional modeling of many practical LSVSs by a Linear Shift-Variant Transform (LSVT). Unlike LSVT, RT is computationally very efficient. It provides an explicit, non-iterative, and closed-form formula for computing the input signal f of an LSVS given the output signal g and a set of moment parameters w of its SV-PSF h. Further, unlike LSVT, RT is naturally suited for local or short-interval or small window signal processing and analysis. RT characterizes signals in a small window using its derivatives which are local features of the signal. Similarly, unlike LSVT, RT provides a computationally efficient method for computing the output signal g of an LSVS given the input signal f and a set of moment parameters w of its SV-PSF h. The method provides a formula that is explicit, non-iterative, in closed-form, and suitable for local signal processing and analysis.
In the computation of input or restored signal f using g and a set of moment parameters w of h, f is obtained as a sum of a set of product terms where each term is the product of a derivative of g with the Inverse Rao Transform (IRT) coefficient S′n of g with respect to h. The IRT coefficients S′n are computed from a set of moment parameters w which comprises the signal domain moments of shift-variance derivatives of h.
Similarly, in the computation of output signal g using fand the set of moment parameters w of h, output signal g is obtained as a sum of a set of product terms where each term is the product of a derivative of f with the forward RT coefficient Sn of input signal f with respect to SV-PSF h. The RT coefficients are computed from the signal domain moments of shift-variance derivatives of h.
In the field of the present invention, computing the input or restored signal f from the output or blurred signal g is considered to be a very difficult and computationally very expensive problem. The present invention addresses this problem and provides a novel method and apparatus for solving this problem. On the other hand, the reverse problem of computing the output or blurred signal g from the input signal f is not considered to be a very hard problem in comparison with the former problem. Nevertheless, the present invention provides a novel method and apparatus for this problem also.
The present invention is also applicable to non-linear systems that can be reformulated or transformed (e.g. through a log transform to convert products to summation) so that they can be modeled by an LSVS. The methods and apparatus disclosed here have applications in many areas of signal and image processing such as medical images, industrial images for inspection and measurement, images captured by many types sensors like digital cameras, microscopes, telescopes, X-ray imaging devices, MRI and radiation based imaging devices, aerial/satellite imagery of the earth, etc.
This invention is relevant to the processing of general multi-dimensional signals including 1-dimensional (1D) time signals, 2-dimensional (2D) image signals, 3-dimensional (3D) video data, 3D X-ray tomographic data, 3D seismographic data, and, in general, n-dimensional signals where n is a positive integer.